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Home/Math Visualization/Conjugate Gradient Solver

Conjugate Gradient Solver

SPD system Ax=b as quadratic minimization: contour geometry, CG vs steepest descent path, residual norm, and condition number.

SPD matrix A

1
14
30°
2

Right-hand side and start

2.2
1.1
-2.2
2

Measured values

Condition number κ14.00
Residual norm ||r||0.00000
Current xk(2.132, 1.219)
Exact x*(2.132, 1.219)

In exact arithmetic, CG reaches the 2D solution in at most two iterations. Extra iterations here mainly show roundoff-scale residuals and how the path differs from steepest descent.

Live graphs

About this model

Conjugate gradient (CG) solves symmetric positive definite systems Ax=b by minimizing the quadratic f(x)=1/2 x^T A x - b^T x. This simulator uses a 2D SPD matrix so the geometry is visible: ellipses are level sets of the quadratic, the exact solution is the ellipse center, the residual is b-Ax, and CG chooses A-conjugate search directions. The same starting point is also shown with steepest descent, making zig-zagging and condition-number effects easy to compare.

Who it's for: Numerical linear algebra, scientific computing, optimization, PDE solvers, and applied math courses.

Key terms

  • Conjugate gradient
  • SPD matrix
  • Residual norm
  • Quadratic minimization
  • Condition number
  • Steepest descent

How it works

Conjugate gradient visualization for a 2D symmetric positive definite system Ax=b, residual norm, and quadratic minimization geometry.

Key equations

CG solves Ax=b by minimizing f(x)=1/2 x^T A x − b^T x for SPD A
α_k=(r_k^T r_k)/(p_k^T A p_k), β_k=(r_{k+1}^T r_{k+1})/(r_k^T r_k)

Frequently asked questions

Why does CG need the matrix to be symmetric positive definite?
SPD makes the quadratic strictly convex and defines the A-inner product used for conjugate directions. Without those properties the classic CG guarantees no longer apply.
Why can CG solve a 2D system in two steps?
In exact arithmetic, CG minimizes over a growing Krylov subspace and terminates in at most n iterations for an n-dimensional SPD system. The 2D demo reaches the solution after at most two independent directions.